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The Clarets had lost 5-0 on their last five visits to City, but made the champions work hard for their victory.
Bernardo Silva and Kevin De Bruyne scored the goals for City in each half, but Burnley created some big chances, only to fail to take any.
Guardiola said: "After the international break, it’s always a little bit difficult to come to a rhythm, and I know how good Burnley are.
”They did really well, they have good players and we played a good 15-20 minutes in the first half, we created chances and the last 20 minutes we lost some tempo.
”We did well in the second half and played really well and could’ve scored more goals.
”Good result, after the international break we take the victory and look what’s next.”
Chris Wood had a big opening late on, after Nathan Collins headed on Dwight McNeil’s free kick, and Guardiola added: "We run and fight for every ball, Burnley play for long balls and they’re a master at this.
”You have to defend set pieces well and anything could’ve happened if Chris Wood had scored that goal near the end.
”We had chances but couldn’t score more unfortunately.
"The importance is improving our game, I am satisfied that the second half was better than first.
”The team showed me the way we want to play and the shape we want to play in.
”All the guys gave everything, it doesn’t matter if the performance is good or not, it’s important that you fight for your teammates and everyone was incredible."